Masatoshi Ohrui

MasatoshiOhrui1993@gmail.com 

Abstract

This is an application of the elementary functional analysis in the sense that there are no long or complicated calculations, and the theory of evolution equations is not used at all. Our initial values can be taken arbitrarily large, our solutions are physically suitable.

Introduction

The existence of the solutions is actually known. For example, Fujita-Kato Theory, Shibata Theory: Kato [8], Zhang [9], Charve-Danchin [10], Shibata-Miura [11]. The semi-group theory or apriori estimates are in these theories, but these are not elementary. The initial values can be taken arbitrary large in the Leray-Hopf's weak solutions, but the uniqueness and smoothness are unresolved. Semi-group theory or apriori estimates are not used in the proof of existence of the Leray-Hopf's weak solutions (for example, Wasao SIBAGAKI, Hisako RIKIMARU [7]), but it is not elementary, too. We define new weak solutions with uniqueness and smoothness, without semi-group theory or apriori estimates. We apply locally solvability of the partial differential oparators with constant coefficients. The policy is, to let L be the heat operator \partial_t-\Delta in the initial value problem of the Navier-Stokes equations on \Omega
\partial_t u -\Delta u=f - \nabla \mathfrak{p}-(u \cdot \nabla)u
\mathrm{div}\,u=0
u(0, x)=a(x),
to erase the pressure \mathfrak{p}, to approximate the nonlinear term (u \cdot \nabla)u by a sequence of smooth functions, to use the locally solvability for the difference between the external force f and the approximation term, and to show that the limit in the Sobolev space is the solution. The uniqueness follows without boundary conditions. Our solutions are physically suitable: \lim_{t, |x|\to\infty}\partial^\alpha u(t, x)=0, and f\mapsto u, f\mapsto\mathfrak{p};a\mapsto u, a\mapsto\mathfrak{p} are continuous.

[Definition of symbols]
For convenience, we write the index of the component of the vectors in the upper right corner. "Function space" and "space" are abbreviations for "linear topological space" (of functions or distributions), other than pressure \mathfrak{p} are \mathbb{R}^3-valued. The absolute value of the functions in the norm of normal function space is interpreted as the length of the number vector (the absolute value of \mathbb{R}^3) in the norm of the space of the \mathbb{R}^3-value functions. We write the space of the real numeric functions and the space of the \mathbb{R}^3-value functions in the same symbol to make symbols simple. Let |\Omega| be \Omega's Lebesgue measure. Let \chi_{\Omega} be the characteristic function on \Omega. For any natural number m \gt \max\{0+4/1, 0+4/2\}=4, p=1, 2, let V^{m, p}(\Omega)=\{ u \in C^{\infty}(\R\times\R^3) : \|u\|_{W^{m, p}(\Omega)} \lt {\infty}\}/(u\sim v:\iff u=v\,\mathrm{on}\,\Omega), V_{\sigma}^{m, p}(\Omega)=\{ u \in C^{\infty}(\R\times\R^3) : \|u\|_{W^{m, p}(\Omega)} \lt {\infty},\mathrm{div}\,u=0\,\mathrm{on}\,\Omega \}/(u\sim v:\iff u=v\,\mathrm{\,on}\,\Omega). Let W^{m, p}(\Omega), W_\sigma^{m, p}(\Omega) be the Sobolev spaces defined by completions: W^{m, p}(\Omega)=\overline{V^{m, p}(\Omega)}^{\| \cdot \|_{W^{m, p}(\Omega)}}, W_{\sigma}^{m, p}(\Omega)=\overline{V_{\sigma}^{m, p}(\Omega)}^{\| \cdot \|_{W^{m, p}(\Omega)}}. Let \mathcal{D}(\Omega) be the space of the test functions (C_{0}^{\infty}(\Omega) as a set), let \mathcal{D}_\sigma(\Omega) be the space of the test functions that the divergence is 0 for spatial variables (see [Supplement 1]). Let P:L^2(\Omega)\to L^2_\sigma(\Omega)=\overline{\mathcal{D}_{\sigma}(\Omega)}^{\| \cdot \|_{L^2(\Omega)}} be the projection. Let C^{k, \varepsilon}(\overline{\Omega}) be the Hölder space. Let
\langle w, \varphi \rangle = (w, \varphi)_{L^2(\Omega)}
=\int_{\Omega} \sum_{i=1}^{3} w^{i}(t, x)\varphi^{i}(t, x)dtdx
=\int_{\Omega} w(t, x) \cdot \varphi(t, x)dtdx
(w=(w^1, w^2, w^ 3), \varphi=(\varphi^1, \varphi^2, \varphi^3)).
In general, if for two Banach spaces X, Y, there exists a linear Hausdorff space Z such that X, Y \subset Z, then X\cap Y is a Banach space with norms given by \|u\|_X+\|u\|_Y or \max\{\|u\|_X, \|u\|_Y\}. \max\{\|u\|_X, \|u\|_Y\}\le \|u\|_X+\|u\|_Y \le 2\max\{\|u\|_X, \|u\|_Y\} so these are equivalent. We put
\mathcal{X}=\bigcap_{m=5}^\infty W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega),
\mathcal{X}'= \bigcap_{m=5}^\infty W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega).
We define for any u\in \mathcal{X},
\|u\|_X=\sum_{m=5}^\infty \frac{1}{m!^5}\|u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)},
for any u\in \mathcal{X}',
\|u\|_{X'}=\sum_{m=5}^\infty \frac{1}{m!^5}\|u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}.
We put X=\{u\in\mathcal{X}:\|u\|_{X}\lt\infty\}, X'=\{u\in\mathcal{X}':\|u\|_{X'}\lt\infty\}.
For a constant M\gt 0, let S be a subset of X:
S=\{u\in X:\|u\|_{X}\le M\}. Let the fundamental solution of \partial_t - \Delta be E. That is, in the sence that \mathbb{R}^3-valued distribution,
(\partial_t - \Delta)E(t, x)=\delta(t, x) = \delta(t) \otimes \delta(x).
Here,
E^{i}(t, x)=\begin{cases} \frac{1}{\sqrt{4 \pi t}^3} e^{-\frac{|x|^2}{4t}} & (t \gt 0) \\ 0 & (t \le 0) \end{cases}([3], [4]).
(END)

Existence of elementary weak solutions

[Assumptions]
Let the domain \Omega be a bounded open set contained in \mathbb{R}\times\mathbb{R}^3, has smooth boundary, and satisfies (0, 0)\in\Omega. Let the external force f:\mathbb{R}\times\mathbb{R}^3\to\R^3 satisfies f\in X' and \|f\|_{X'}\le M^2. If f\neq 0 then we let \int_{\mathbb{R}\times \mathbb{R}^3} E(s, y) \, \chi_{\Omega}(\cdot-s, \cdot-y)Pf(\cdot-s, \cdot-y)dsdy\neq 0.
Let the set of initial values be
A=\{u(0, \cdot):u \in S, u(t, x)=\int_{\mathbb{R}\times \mathbb{R}^3} E(s, y) \, \chi_{\Omega}(t-s, x-y)(\,Pf(t-s, x-y) - P((u\cdot \nabla)u)(t-s, x-y))dsdy\}.
(END)

Prop. 0. [Locally solvability of linear partial differential operator with constant coefficients]
Let the fundamental solution of linear partial differential operator with constant coefficients L on \mathbb{R}^N be E. E \in \mathcal{D}^{\prime} satisfies LE=\delta. For f \in L^1_{\mathrm{loc}}, one of the weak solutions of the equation Lu=f on \Omega\Subset\mathbb{R}^N is u=E * \chi_\Omega f \in \mathcal{D}^{\prime}(\Omega).
(END)

[Proof]
For any \varphi\in\mathcal{D}(\Omega),
\langle L(E*\chi_\Omega f), \varphi\rangle
=\pm\langle E*\chi_\Omega f, L\varphi\rangle
:=\pm\langle E(x), \langle \chi_\Omega(y)f(y), L\varphi(x+y)\rangle\rangle
= \pm\langle \chi_\Omega(x)f(x), \langle E(y), L\varphi(x+y)\rangle\rangle
= \langle \chi_\Omega(x)f(x), \langle LE(y), \varphi(x+y)\rangle\rangle
=\langle LE(x), \langle \chi_\Omega(y)f(y), \varphi(x+y)\rangle\rangle
=\langle \chi_\Omega(y)f(y), \varphi(y)\rangle =\langle f, \varphi\rangle.
(END)

Prop. 1. [Main theorem]
A\neq\empty. Let a\in A. Then there are elementary weak solutions u, \mathfrak{p} of the initial value problem on \Omega:

\partial_t u -\Delta u=f - \nabla \mathfrak{p}-(u \cdot \nabla)u
\mathrm{div}\,u=0
u(0, x)=a(x),

in the sence that, u \in S, \mathfrak{p}\in L_{\mathrm{loc}}^2(\Omega)/(\mathfrak{p}'\sim\mathfrak{q}\iff\nabla\mathfrak{(p'-q)}=0) and u, \mathfrak{p} satisfy for any \varphi \in \mathcal{D}_\sigma(\Omega), \langle \partial_t u + (u \cdot \nabla)u - \Delta u + \nabla \mathfrak{p} - f, \varphi \rangle =0,
for any \varphi\in\mathcal{D}(\Omega),
\langle\mathrm{div} \,u, \varphi\rangle=-\sum_{j=1}^3\langle u^j, \partial_{x^j}\varphi\rangle=0.

If f \neq 0 then A\notni 0, u \neq 0; if f=0 then A=\{0\}, u=0. For any multi-index \alpha,
\lim_{t, |x|\to\infty}\partial^\alpha u(t, x)=0.
If f satisfies for any c\in X,
\limsup_{t, |x|\to\infty}f(t, x)\neq \limsup_{t, |x|\to\infty}(\partial_t c(t, x) -\Delta c(t, x)+(c \cdot \nabla)c(t, x)), then the solutions u,\mathfrak{p} and v,\mathfrak{q} satisfy u=v, \mathfrak{p}=\mathfrak{q}\,\mathrm{on}\, \Omega. f\mapsto u, f\mapsto\mathfrak{p};a\mapsto u, a\mapsto\mathfrak{p} are continuous.
(END)

We proof it later by Banach's fixed point theorem.

Prop. 2. [Smoothness of elementary weak solutions]
Solution (u, \mathfrak{p}) are C^{\infty}-functions.
(END)

[Proof of smoothness]
m can be arbitrarily large, so the embedding theorem into Hölder space ([5]), 
"if \mathbb{N}\ni m-4/p\gt 0 then W^{m, p}(\Omega)\subset C^{(m-4/p)-1, \varepsilon}(\overline{\Omega}) for \varepsilon\in (0, 1)", in the sence of existence of suitable representative elements, u is C^\infty-function.

f is smooth and \partial_t u + (u \cdot \nabla)u - \Delta u - f=-\nabla \mathfrak{p}. Because -\nabla \mathfrak{p} is smooth, so \mathfrak{p} is also smooth. (END)

Lemma 0.
\chi_{\Omega}\in X, \chi_{\Omega}\in X' so X, X'\neq \{0\}. X, X' are norm spaces.
(END)

Lemma 1. [Completeness]
X, X' are Banach spaces.
(END)

[Proof]
Let \{u_n\} be a Cauchy sequence in X. Then, for any m\ge 5, \{u_n\} is a Caucy sequence of W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega). W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega) is a Banach space, so \{u_n\} converges. Let the limit be u. u\in X. For any positive number \varepsilon, there exists a natural number N such that if \ell, n\ge N then
\|u_\ell-u_n\|_X \lt \varepsilon.
From using Fatou's lemma for counting measure,
\|u-u_n\|_X
=\sum_{m=5}^{\infty} \frac{1}{m!^5}\|u-u_n\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}
= \sum_{m=5}^{\infty} \frac{1}{m!^5}\liminf_{\ell\to\infty}\|u_\ell-u_n\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}
\le\liminf_{\ell\to\infty} \sum_{m=5}^{\infty} \frac{1}{m!^5}\|u_\ell-u_n\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}
\le \varepsilon.
(END)

Lemma 2. [Separation of product]
Constant C_1\gt 0 exists such that
\left\|u^i v^i\right\|_{X'}\le C_1\|u^i\|_{X'}\|v^i\|_{X'}
holds for any u, v\in X'.
(END)

[Proof]
For the binomial coefficients c_{\alpha, \beta}, let
c_{\alpha}=\sum_{\beta\le\alpha}c_{\alpha, \beta}.
There is a continuous embedding X'\subset C^{k, \varepsilon}(\overline{\Omega}) for any natural number k, because \|u_n-u\|_X\to 0
\Rightarrow \|u_n-u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)} \to 0
\Rightarrow \|u_n-u\|_{C^{k, \varepsilon}(\overline{\Omega})}\to 0([5]), so there exists a constant c'\gt 0 such that
\|u\|_{C^{k, \varepsilon}(\overline{\Omega})}\le c'\|u\|_{X'}.
If |\alpha|\le k, by Leibniz' formula,
\|\partial^\alpha (u^i v^i)\|_{L^p(\Omega)}
\le c_{\alpha}\|u^i\|_{C^{k, \varepsilon}(\overline{\Omega})}\|v^i\|_{C^{k, \varepsilon}(\overline{\Omega})}|\Omega|^{1/p}
\le c_{\alpha}c' |\Omega|^{1/p}\|u^i\|_{X'} c'\|v^i\|_{X'}
\le c_{\alpha}c'^2 |\Omega|^{1/p}\|u^i\|_{X'}\|v^i\|_{X'}. Therefore,
\|\partial^\alpha (u^i v^i)\|_{L^p(\Omega)}\le c_{\alpha}c'^2 |\Omega|^{1/p}\|u^i\|_{X'}\|v^i\|_{X'}, so there exists a constant C_1\gt 0 such that
\|u^i v^i\|_{X'}\le C_1\|u^i\|_{X'}\|v^i\|_{X'}.
(END)

Lemma 3. [Absorption of differential]
Constant C_2\gt 0 exists such that
\left\|\partial_{x^j}u\right\|_{X}\le C_2\|u\|_{X}
holds for any u\in X.
(END)

[Proof]
Let \{u_n\}\subset X satisfies u_n\to u, \partial_{x^j}u_n\to v. From Hölder's inequality ([2]), we have
|\langle \partial_{x^j}u_n - v, \varphi\rangle|
\le \|\partial_{x^j}u_n - v\|_{L^p(\Omega)}\|\varphi\|_{L^q(\Omega)}
\to 0\,(p=1\Rightarrow q=\infty, p=2\Rightarrow q=2) and the weak differentiation is continuous in \mathcal{D}'_\sigma(\Omega), so \partial_{x^j}u_n\to \partial_{x^j}u\, \mathrm{in}\,\mathcal{D}'_\sigma(\Omega). From
v=\partial_{x^j}u\in X, \{u\in X:\partial_{x^j}u\in X\}=X, the absorption of differential is true by the closed graph theorem.

Or
\|\partial_{x^j}u\|_{X}= \sum_{m=5}^{\infty} \frac{1}{m!^5}\|\partial_{x^j}u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}
\le\sum_{m=4}^{\infty} \frac{1}{m!^5}\|u\|_{W^{m+1, 1}(\Omega)\cap W^{m+1, 2}(\Omega)}
\le C_2\sum_{m=5}^{\infty} \frac{1}{m!^5}\|u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}.
(END)

Lemma 4. [Boundness of X\ni u\mapsto E*(\chi_\Omega u)\in X]
X\ni u\mapsto E*(\chi_\Omega u)\in X is a bounded operator, so constant C_3\gt 0 exists such that for any u\in X,
\|\int_{\mathbb{R}\times\mathbb{R}^3}E(s, y)\chi_\Omega(t-s, x-y)u(t-s, x-y)dsdy\|_{X}
\le C_3\|u\|_X
holds.
(END)

[Proof]
As a function of (s, y), for almost every (t, x)\in\Omega,
\mathrm{supp}(E^i(s, y) \chi_\Omega(t-s, x-y)u^i(t-s, x-y))
\subseteq -\overline{\Omega}+(t, x)
=\overline{\{(s, y)\in \mathbb{R}\times\mathbb{R}^3:(t-s, x-y)\in\Omega\}}
is the translation of reverse of \overline{\Omega}, so it is compact, and
|\partial_{t, x}^\alpha(E^i(s, y) \chi_\Omega(t-s, x-y)u^i(t-s, x-y))|\le E^i(s, y)\sup\{|\partial_{t, x}^\alpha u^i(t-s, x-y)|:(t, x)\in\Omega\}\le C_\alpha E^i(s, y)\in L^1_{s, y}(\Omega), so combining the theorem of differentiation under the integral sign, Hölder's inequality and continues embedding X\subset L^\infty(\Omega), we have
\|\partial^\alpha(E*(\chi_{\Omega}u))\|_{L^p(\Omega)}
\le\|E*(\partial^\alpha (\chi_{\Omega}u))\|_{L^p(\Omega)}
\le \|\|E(s, y)\|_{L_{s, y}^1(-\Omega+(t, x))}\|\partial^\alpha u(t-s, x-y)\|_{L_{s, y}^\infty(-\Omega+(t, x))}\|_{L_{t, x}^p(\Omega)}
\le \sup\{\|E\|_{L^1(-\Omega+(t, x))}:(t, x)\in\Omega\}\|\partial^\alpha u\|_{L^\infty(\Omega)}|\Omega|^{1/p}
\le \sup\{\|E\|_{L^1(-\Omega+(t, x))}:(t, x)\in\Omega\}c''C_2^{|\alpha|}\|u\|_X|\Omega|^{1/p}
\lt\infty.
So we have
\|E*(\chi_\Omega u)\|_X\le C_3\|u\|_X.
(END)

We take C=\max\{C_1, C_2, C_3\}. Lemma 2, 3, 4 hold for C. We take M\gt 0 satisfying C(1+3C^2)M\le 1.

We solve
(N-S)'\partial_t u -\Delta u=f -(u \cdot \nabla)u,
that is, for any a\in A, there exist u \in S, \mathfrak{p}\in L_{\mathrm{loc}}^2(\Omega)/(\mathfrak{p}'\sim\mathfrak{q}\iff\nabla\mathfrak{(p'-q)}=0), such that for any \varphi \in \mathcal{D}_\sigma(\Omega),
\langle \partial_t u + (u \cdot \nabla)u - \Delta u + \nabla \mathfrak{p} - f, \varphi \rangle =0,
for any \varphi\in\mathcal{D}(\Omega),
\langle\mathrm{div} \,u, \varphi\rangle=-\sum_{j=1}^3\langle u^j, \partial_{x^j}\varphi\rangle=0,
u(0, x)=a(x).

\varPhi:S\to S can be defined as
\varPhi[u](t, x)
=\int_{\mathbb{R}\times\mathbb{R}^3} E(s, y)\chi_{\Omega}(t-s, x-y)(Pf(t-s, x-y) -P((u\cdot\nabla)u)(t-s, x-y))dsdy. We take a function sequence \{u_n\}\subset S as u_0\in S, if n\ge 0 then u_{n+1}(t, x)=\varPhi[u_n](t, x)
=\int_{\mathbb{R}\times\mathbb{R}^3} E(s, y) \chi_{\Omega}(t-s, x-y)(Pf(t-s, x-y) -P((u_n\cdot\nabla)u_n)(t-s, x-y))dsdy. From Lemma 0, 1, X is a complete metric space, so S is complete because it is a closed subset that is not empty, and if \varPhi is a contraction mapping, according to the Banach's fixed point theorem ([1], [6]), the uniqueness and the existence of a fixed point of \varPhi follows:

Some u \in S exists uniquely and \varPhi[u]=u.

Then, due to the uniqueness of the fixed point in Banach's fixed point theorem, u is an unique weak solution. If f \neq 0 then A\notni 0, u \neq 0. Let f=0. From the properties of X, if u\in X and u(t, x)=\int_{\mathbb{R}\times\mathbb{R}^3} E(s, y)\chi_{\Omega}(t-s, x-y)(Pf(t-s, x-y) -P((u\cdot\nabla)u)(t-s, x-y))dsdy then \|u\|_X\le 3C^3\|u\|_X^2. So, if u\neq 0 then 1\le 3C^3\|u\|_X. By C=O(|\Omega|)\,(|\Omega|\to 0) and the absolute continuity of Lebesgue integral, |\Omega|\to 0\Rightarrow C\to 0, \|u\|_X\to 0, therefore f=0\Rightarrow A=\{0\}, u=0.

Lemma 5. [Possibility that \varPhi can be defined as a contraction mapping]
u\in S\Rightarrow \|E*(\chi_{\Omega}(Pf-P((u\cdot \nabla)u)))\|_X\lt\infty
holds. Therefore
\|\varPhi[u]\|_X\le M.
(END)

[Proof]
\|P\|=1, so
\|\chi_{\Omega}(Pf-P((u\cdot\nabla)u))\|_X
\le\|f\|_{X'}+\|u^1 \partial_{x^1}u+u^2 \partial_{x^2}u+u^3 \partial_{x^3}u\|_{X'}
\le M^2+3C^2M^2\lt\infty.

If
\|\varPhi[u]\|_X
\le CM^2+3C^3M^2
\le M, M must be C(1+3C^2)M\le 1.
(END)

Lemma 6. [\varPhi is a contraction mapping]
\varPhi:S\to S is Lipschitz continuous: there is a constant L\gt 0 such that \|\int_{\mathbb{R}\times\mathbb{R}^3}E(s, y) \chi_{\Omega}(t-s, x-y)(P((v \cdot \nabla)v)(t-s, x-y)-P((u\cdot\nabla)u)(t-s, x-y))dsdy\|_X
\le L \|u- v\|_X,
L \lt 1.
(END)

[Proof]
(v \cdot \nabla)v(t-s, x-y)-(u \cdot \nabla)u(t-s, x-y)
=\sum_{j=1}^3 (v^j (\partial_{x^j}v(t-s, x-y) - \partial_{x^j}u(t-s, x-y)) + (v^j \partial_{x^j}u(t-s, x-y)) - (u^j \partial_{x^j}u(t-s, x-y))), so we have

\|\int_{\mathbb{R}\times\mathbb{R}^3}E(s, y) \chi_{\Omega}(t-s, x-y)(P((v \cdot \nabla)v)(t-s, x-y)-P((u\cdot\nabla)u)(t-s, x-y))dsdy\|_X
\le C^2\|v\|_X\max_j(\|\partial_{x^j}v - \partial_{x^j}u\|_X)+C^2\|v-u\|_X\max_j(\|\partial_{x^j}u\|_X)
\le C^3M\|v-u\|_X+C^3M\|v-u\|_X
= 2C^3M\|u- v\|_X.

Therefore, Lipschitz continuity follows for L=2C^3M.

From the above argment
\|\int_{\mathbb{R}\times\mathbb{R}^3}E(s, y)\chi_{\Omega}(t-s, x-y)(P((v \cdot \nabla) v(t-s, x-y))-P((u \cdot \nabla)u)(t-s, x-y))dsdy\|_X
\le 2C^3M\|u- v\|_X
and
2C^3M\lt 1.
(END)

Lemma 7.
For any U\in X,
" \varphi\in\mathcal{D}_\sigma(\Omega)\Rightarrow\langle U, \varphi\rangle =0 "
\iff " there exists a distribution \mathfrak{p} such that U=\nabla\mathfrak{p} ".
(END)

[Proof]
For any \varphi\in\mathcal{D}_\sigma(\Omega),
\mathrm{div}(\varphi)=0, so by the integration by parts
\langle \nabla\mathfrak{p}, \varphi\rangle
=\int_{\Omega} \sum_{i=1}^{3} (\nabla\mathfrak{p})^i(t, x)\varphi^i(t, x)dtdx
=-\int_{\Omega}\mathfrak{p}(t, x)\mathrm{div}(\varphi)(t, x)dtdx=0.

So by Helmholtz decomposition, if
U=PU+\nabla\mathfrak{p}
then
\langle U,\varphi\rangle=\langle\nabla\mathfrak{p},\varphi\rangle=0.
(END)

Lemma 8. [Solvability of the Navier-Stokes equations]
The fixed point u of \varPhi:S\to S is the solution of (N-S)'.
(END)

[Proof]
u satisfies \mathrm{div}\,u=0 in the sense of a distribution belonging to \mathcal{D}'(\Omega). That is, for any \varphi\in\mathcal{D}(\Omega), \langle\mathrm{div}\,u, \varphi\rangle=-\sum_{j=1}^3\langle u^j, \partial_{x^j}\varphi\rangle=0.
In fact, for any u\in W_\sigma^{m, p}(\Omega) there exists a Cauchy sequence \{u_n\}\subset V_\sigma^{m, p}(\Omega), by the integration by parts and Hölder's inequality, we have
0=-\sum_{j=1}^3\langle u_n^j, \partial_{x^j}\varphi\rangle
\to -\sum_{j=1}^3\langle u^j, \partial_{x^j}\varphi\rangle.

Boundness of u, \partial_{x^j}u by Sobolev embedding theorem and |\Omega|\lt\infty, we have (u\cdot\nabla)u\in L^2(\Omega), so by Helmholtz decomposition,
if we let f=Pf+\nabla\mathfrak{f}, (u\cdot\nabla)u=P((u\cdot\nabla)u)+\nabla\mathfrak{u}
then for any \varphi\in\mathcal{D}_\sigma(\Omega),
\langle f, \varphi\rangle = \langle Pf, \varphi\rangle, \langle (u\cdot\nabla)u, \varphi\rangle =\langle P((u\cdot\nabla)u), \varphi\rangle, hence we solve

(N-S)' \partial_t u - \Delta u= f -(u \cdot \nabla)u\,\mathrm{in}\, \mathcal{D}'_\sigma(\Omega).

By Prop. 0, the solution of the approximate equation on \Omega

(N-S)'' \partial_t v_{n} - \Delta v_{n} =Pf-P((u_n \cdot \nabla)u_n)
satisfies
v_n=u_{n+1}=E * \chi_{\Omega}(Pf -P((u_n \cdot \nabla)u_n)).

Therefore, the solution of (N-S)'' satisfies
u_{n+1}(t, x)=\int_{\mathbb{R} \times \mathbb{R}^3} E(s, y) \chi_{\Omega}(t-s , x-y)(Pf(t-s, x-y) -P((u_n \cdot \nabla)u_n)(t-s, x-y))dsdy.

\partial_t u_{n+1} (t, x)- \Delta u_{n+1} (t, x)
=\langle(\partial_t E(t-s, x-y) - \Delta E(t-s, x-y)),\chi_{\Omega}(s, y)(Pf(s, y)-P((u_n \cdot \nabla)u_n)(s, y))\rangle
=\langle\delta(\tau) \otimes \delta(z),\chi_{\Omega}(t-\tau, x-z)(Pf(t-\tau, x-z)-P((u_n \cdot \nabla)u_n)(t-\tau, x-z)) \rangle
=Pf(t, x)-P((u_n \cdot \nabla)u_n)(t, x).

From Hölder's inequality, \|P\|=1, and the continuity of product of the functions L^2(\Omega)\times L^2(\Omega) \ni (u, v) \mapsto uv \in L^1(\Omega) (see [Supplement 2]), we have
| \int_{\Omega} (P((u_n \cdot \nabla)u_n)(t, x)
-P((u \cdot \nabla)u)(t, x))) \cdot \varphi(t, x) dtdx |
\le \|((u_n \cdot \nabla)u_n)(t, x)-((u \cdot \nabla)u)(t, x)\|_{L^1(\Omega)}\| \varphi(t, x) \|_{L^\infty(\Omega)}\to 0\,(n \to \infty). From continuity of the heat operator on \mathcal{D}'_\sigma(\Omega):
|\langle \partial_t u_{n+1} - \Delta u_{n+1}, \varphi \rangle - \langle \partial_t u - \Delta u, \varphi \rangle|\to 0, 
\partial_t u - \Delta u =Pf-P((u \cdot \nabla)u) holds. From Lemma 5, 6, we have
u(t, x)=\int_{\mathbb{R} \times \mathbb{R}^3} E(s, y)\chi_{\Omega}(t-s, x-y) (Pf(t-s, x-y)-P((u \cdot \nabla)u)(t-s, x-y))dsdy.

u is a solution in the sence of distribution in \mathcal{D}_\sigma'(\Omega) of (N-S)' .

From Lemma 7, there exists \mathfrak{p} such that \partial_t u + (u \cdot \nabla)u - \Delta u - f=-\nabla \mathfrak{p} holds.
(END)

Properties of the solutions

Lemma 9. [Vanishing]
\lim_{t, |x|\to\infty}\partial^\alpha u(t, x)=0.
(END)

[Proof]
u is a measurable function on \R\times\R^3, so we can take the limits as t, |x|\to\infty.

\partial^\alpha u(t, x)=\int_{\Omega} E(t-s, x-y) \, \partial^\alpha(Pf(s, y) - P((u\cdot \nabla)u)(s, y))dsdy, for any t_0\gt 0, if t-s\gt t_0 then
|E^i(t-s, x-y)|\le 1/t_0^{3/2},
\partial^\alpha(Pf- P((u\cdot \nabla)u))\in X\subset C^{0, \varepsilon}(\overline{\Omega})
so \lim_{t, |x|\to\infty}\partial^\alpha u(t, x)=0 follows from the bounded convergence theorem.
(END)

Lemma 10. [Uniqueness]
If f satisfies for any c\in X,
\limsup_{t, |x|\to\infty}f(t, x)\neq \limsup_{t, |x|\to\infty}(\partial_t c(t, x) -\Delta c(t, x)+(c \cdot \nabla)c(t, x)), then the solutions u,\mathfrak{p} and v,\mathfrak{q} satisfy u=v, \mathfrak{p}=\mathfrak{q}\,\mathrm{on}\, \Omega.
(END)

[Proof]
If u\neq v, there exists c\in X such that u=v+c, c\neq 0.
\partial_t (v+c) -\Delta (v+c)+((v+c) \cdot \nabla)(v+c)=f.
From Lemma 9,
\limsup_{t, |x|\to\infty}f(t, x)=\limsup_{t, |x|\to\infty}(\partial_t c(t, x) -\Delta c(t, x)+(c \cdot \nabla)c(t, x)).
This is a contradiction.
(END)

Lemma 11. [Continuity of f\mapsto u, f\mapsto\mathfrak{p}]
Let f_n, f\in X', \|f_n\|_{X'}, \|f\|_{X'}\le M^2, \|f_n-f\|_{X'}\to 0. Let the solutions be u_n,\mathfrak{p}_n for f_n and a_n\in A, let the solutions be u,\mathfrak{p} for f and a\in A. Then
\|u_n-u\|_X\to 0,
d(\mathfrak{p}_n, \mathfrak{p}):=\|u_n-u\|_X\to 0.
(END)

[Proof]
\|u_n-u\|_X=\|\int_{\mathbb{R}\times \mathbb{R}^3} E(s, y) \, \chi_{\Omega}(t-s, x-y)(\,Pf_n(t-s, x-y) - P((u_n\cdot \nabla)u_n)(t-s, x-y))dsdy-\int_{\mathbb{R}\times \mathbb{R}^3} E(s, y) \, \chi_{\Omega}(t-s, x-y)(\,Pf(t-s, x-y) - P((u\cdot \nabla)u)(t-s, x-y))dsdy\|_X
\le C\|f_n-f\|_X+2C^3M\|u_n-u\|_X.
So
\limsup_{n\to\infty}\|u_n-u\|_X
\le 2C^3M\limsup_{n\to\infty}\|u_n-u\|_X.

\limsup_{n\to\infty}\|u_n-u\|_X\le 2M,
therefore
0\le (1-2C^3M)\limsup_{n\to\infty}\|u_n-u\|_X
\le 0.
Hence
\limsup_{n\to\infty}\|u_n-u\|_X
=\lim_{n\to\infty}\|u_n-u\|_X=0.
There exist the maps f\mapsto u, u\mapsto\mathfrak{p} so d(\mathfrak{p}_n, \mathfrak{p}):=\|u_n-u\|_X\to 0.
(END)

Lemma 12. [Continuity of a\mapsto u, a\mapsto\mathfrak{p}]
Let the solutions be u_a, v_b, \mathfrak{p}_a, \mathfrak{q}_b for a, b. If we define the metrics given by
d_A(a, b)=\|u_a-v_b\|_X, D(\mathfrak{p}, \mathfrak{q})=\|u_a-v_b\|_X, then a\mapsto u, a\mapsto\mathfrak{p} are continuous.
(END)

Prop. 1 follows from Lemma 0, …, 12.

From the properties of X and u=\varPhi[u],
\|u\|_X\le C\|f\|_{X'} +3C^3\|u\|_X^2\le M.
CM\le C(1+3C^2)M\le 1
so
C\|f\|_{X'}\le CM^2\le M.
Therefore, from
C\|f\|_{X'} +3C^3\|u\|_X^2\le M, we have
\|u\|_X\le \sqrt{\frac{M-C\|f\|_{X'}}{3C^3}}\lt M.

[Elementary weak solutions as Bochner class]
Let
I=\{t\in\R:\exists{x}\in\R^3, (t, x)\in\Omega\},
\Omega'=\{x\in\R^3:\exists{t}\in\R, (t, x)\in\Omega\}.
For 1\le p\lt\infty,
a\in C^\infty(\overline{\Omega'})\cap L^p_\sigma(\Omega'), u\in C(I;L^p_\sigma(\Omega')).
(END)

[Proof]
u\in C(\Omega), so for any \varepsilon\gt 0, there exists a \delta\gt 0 such that
(t, x), (t', x)\in\Omega, |(t, x)-(t', x)|\lt\delta
\Rightarrow |u(t, x)-u(t', x)|\lt\varepsilon.
Therefore
\|u(t, \cdot)-u(t',\cdot)\|_{L^p_\sigma(\Omega')}\le |\Omega'|^{1/p}\varepsilon.
(END)

Supplements

[Supplement 1]
As functions \varphi that \mathrm{div} \varphi = \nabla \cdot \varphi=0, it is sufficient to take any \psi \in \mathcal{D}(\Omega) and set to \varphi = \mathrm{curl} \psi.

[Supplement 2]
Let \|u_n-u\|_{L^2(\Omega)}\to 0, \|v_n-v\|_{L^2(\Omega)}\to 0. By the triangle inequality, we have
| \|u_n\|_{L^2(\Omega)}-\|u\|_{L^2(\Omega)}|\le \|u_n-u\|_{L^2(\Omega)} for any sufficientaly large n. On the other hand, \|u_n\|_{L^2(\Omega)}\lt \|u\|_{L^2(\Omega)}+1. Therefore
\|u_n v_n - uv\|_{L^1(\Omega)}\le \|u_n\|_{L^2(\Omega)}\|v_n-v\|_{L^2(\Omega)}+\|v\|_{L^2(\Omega)}\|u_n-u\|_{L^2(\Omega)}
\lt (\|u\|_{L^2(\Omega)}+1)\|v_n-v\|_{L^2(\Omega)}+\|v\|_{L^2(\Omega)}\|u_n-u\|_{L^2(\Omega)} \to 0.

[Supplement 3]
We change the assumptions: I\ni 0, \Omega'\notni 0. Let \rho(0, \cdot)\in X'(\Omega').

We can solve
\partial_t \rho +\mathrm{div}(\rho u)=0
\mathrm{Div}(D(u)+\mathrm{div}u(\delta^{ij})-\mathfrak{p}(\delta^{ij}))=f-\rho(\partial_t u+(u\cdot\nabla)u)
similarly. Here,
(\mathrm{Div}T)^i=\sum_{j=1}^3\partial_{x^j}T^{ij}, D^{ij}(u)=\partial_{x^j}u^i+\partial_{x^i}u^j.

An operator L_u:X'\ni\rho\mapsto \mathrm{div}(\rho u)\in X' is bounded, so
\rho=e^{-tL_u}\rho(0,\cdot).
Then
\mathrm{Div}(D(u)+\mathrm{div}u(\delta^{ij})-\mathfrak{p}(\delta^{ij}))=f-e^{-tL_u}\rho(0,\cdot)(\partial_t u+(u\cdot\nabla)u).
The fundamental solutions of the elliptic operators are locally integrable on new \Omega, so using above lemmas, the uniquely existence, smoothness and properties of the solutions follow similarly. The initial velocity vectors can be arbitrarily large.

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